possible only for real or complex solutions, and consists of outputting numeric approximations of the solutions. Such a solution is called numeric. A solution is certified if it is provided with a bound on the error of the approximations, and if this bound separates the different solutions. The other way of representing the solutions is said to be algebraic. It uses the fact that, for a zero-dimensional system, the solutions belong to the algebraic closure of the field k of the coefficients of the system. There are several ways to represent the solution in an algebraic closure, which are discussed below. All of them allow one to compute a numerical approximation of the solutions by solving one or several univariate equations. For this computation, it is preferable to use a representation that involves solving only one univariate polynomial per solution, because computing the roots of a polynomial which has approximate coefficients is a highly unstable problem. Extensions Trigonometric equations A trigonometric equation is an equation where is a trigonometric polynomial. Such an equation may be converted into a polynomial system by expanding the sines and cosines in it (using sum and difference formulas), replacing and by two new variables and and adding the new equation . For example, because of the identity solving the equation is equivalent to solving the polynomial system For each solution of this system, there is a unique solution of the equation such that . In the case of this simple example, it may be unclear whether the system is, or not, easier to solve than the equation. On more complicated examples, one lacks systematic methods for solving directly the equation, while software are available for automatically solving the corresponding system. Solutions in a finite field When solving a system over a finite field with elements, one is primarily interested in the solutions in . As the elements of are exactly the solutions of the equation , it suffices, for restricting the solutions to , to add the equation for each variable . Coefficients in a number field or in a finite field with non-prime order The elements of an algebraic number field are usually represented as polynomials in a generator of the field which satisfies some univariate polynomial equation. To work with a polynomial system whose coefficients belong to a number field, it suffices to consider this generator as a new variable and to add